Reflection principle

Figure 5.7 Scheaatic dlagraa of refractive index detectors eaploying the principle of refraction (deflection-refractoseter), A, and the reflection principle (Presnel-refractoaeter), B.

Total immersion thermometers, 24 464 Total internal reflection principle,... 

Exercise. For the infinite symmetric random walk an explicit solution of an absorbing boundary problem can be obtained by the reflection principle. Solve the M-equation with initial condition p (0) = 5wm — 5w m. The solution pn(t) for n> 0 obeys... 

Only a few problems with artificial boundaries can be treated by the reflection principle. In this section the method of normal modes is expounded, which in principle is able to deal with artificial boundaries of all types. Rather than develop this method in full generality we demonstrate it on the example (ii) of section 7 the model for diffusion-controlled chemical reactions. 

Exercise. Show that, in contrast, the average time corresponding to (8.14) is infinite. Exercise. Solve the absorbing boundary problem (7.6). Show that for c = 1 the solution coincides with the one found by the reflection principle (7.12). 

The photo-dissociation dynamics at 193 nm was analyzed in detail and the observed rotational state distribution was obtained by using the rotation reflection principle by Schinke and Stasemler [53]. All rotational state distributions depend sensitively on the anisotropy of the dissociative potential energy surface. These are interpreted as a mapping of the bound state wave function onto the quantum number axis. The mapping is mediated by the classical excitation function determined by running classical trajectories onto the potential energy surface within the dissociative state. This so-called rotation-reflection principle... 

Fig. 3.5. Adiabatic potential curves en(R), defined in (3.31), for the model system illustrated in Figure 2.3. The right-hand side depicts three selected partial photo dissociation cross sections cr(Ef,n) for the vibrational states n = 0 (short dashes), n = 2 (long dashes), and n = 4 (long and short dashes). The vertical and the horizontal arrows illustrate the reflection principle (see Chapter 6). Also shown is the total cross section (Jtot Ef) ...

We assume the ground-state potential to be harmonic and the parent molecule to be in the lowest vibrational state. Despite its simplicity, we shall describe the one-dimensional reflection principle in detail because the subsequent extension to more than one dimension follows along the same lines. 

Fig. 6.1. Schematic illustration of the one-dimensional reflection principle. The solid curve on the right-hand side shows the spectrum for a linear potential whereas the dashed curve represents a more realistic case. Ve is the vertical energy defined as V(Re).

A linear approximation of the potential is certainly too sweeping a simplification. In reality, Vr varies with the internuclear separation and usually rises considerably at short distances. This disturbs the perfect (mirror) reflection in such a way that the blue side of the spectrum (E > Ve) is amplified at the expense of the red side (E < 14).t For a general, nonlinear potential one should use Equations (6.3) and (6.4) instead of (6.6) for an accurate calculation of the spectrum. The reflection principle is well known in spectroscopy (Herzberg 1950 ch.VII Tellinghuisen 1987) the review article of Tellinghuisen (1985) provides a comprehensive list of references. For a semiclassical analysis of bound-free transition matrix elements see Child (1980, 1991 ch.5), for example. 

Fig. 6.3. Schematic illustration of the two-dimensional reflection principle and the dependence of the width of the spectrum on the gradients dV/dR and dV/dr. The dots indicate the two different excitation points. The vertical energies Ve = V(Re,re) are different in the two cases and therefore the spectra Eire plotted as functions of E — Ve rather than E.

The energy dependence of each one-dimensional cross section g(E, n) can be easily explained by the one-dimensional reflection principle as illustrated in Figure 6.4. The vertical energy en(Re) determines the peak... 

Fig. 6.4. Schematic illustration of the multi-dimensional reflection principle in the adiabatic limit. The left-hand side shows the vibrationally adiabatic potential curves en(R). The independent part of the bound-state wavefunction in the ground electronic state is denoted by

The reflection principle, outlined in Sections 6.1 and 6.2, explains the energy dependence of absorption spectrum as a mapping of the initial coordinate distribution in the electronic ground state onto the energy axis. Rotational state distributions of diatomic photofragments in direct dissociation can be explained in an analogous manner. 

Equation (6.27) manifests the rotational reflection principle (Schinke 1986c Schinke and Engel 1986) as illustrated in Figure 6.6 ... 

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